Eugene must take four exams in a geography class. If his scores on the first three exams are 91, 67, and 82, what score does he need on the fourth exam for his overall mean to be the following? (a) at least 70 (b) at least 80
Eugene must take four exams in a geography class. If his scores on the first three exams are 91, 67, and 82, what score does he need on the fourth exam for his overall mean to be the following? (a) at least 70 (b) at least 80
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![**Question: Determining Required Scores for Desired Mean in Geography Exams**
Eugene must take four exams in a geography class. If his scores on the first three exams are 91, 67, and 82, what score does he need on the fourth exam for his overall mean to be the following?
(a) at least 70 ______
(b) at least 80 ______
---
**Explanation for Students:**
To solve this problem, let's understand what mean (or average) is first. The mean of a set of numbers is the sum of those numbers divided by the quantity of numbers.
Here's a step-by-step method to solve this:
1. Sum of the first three exam scores:
\[ 91 + 67 + 82 = 240 \]
2. Let the fourth exam score be denoted as \( X \).
3. For part (a), we need the mean of all four exams to be at least 70.
4. Write down the equation for the mean of four exams:
\[ \text{Mean} = \frac{\text{Sum of 4 exam scores}}{\text{Number of exams}} \]
\[ 70 \leq \frac{240 + X}{4} \]
5. Multiply both sides by 4 to clear the fraction:
\[ 70 \times 4 \leq 240 + X \]
\[ 280 \leq 240 + X \]
6. Solve for \( X \):
\[ X \geq 280 - 240 \]
\[ X \geq 40 \]
So, Eugene needs at least a score of 40 on the fourth exam to have an average of at least 70.
Next, let's work on part (b):
1. For part (b), we need the mean of all four exams to be at least 80.
2. Use the same step-by-step method to write down the equation:
\[ 80 \leq \frac{240 + X}{4} \]
3. Multiply both sides by 4:
\[ 80 \times 4 \leq 240 + X \]
\[ 320 \leq 240 + X \]
4. Solve for \( X \):
\[ X \geq 320 - 240 \]
\[ X \geq 80 \]
So, Eugene needs at least a score of 80 on the fourth exam to have an average of at least 80](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F71316fb1-8245-4500-b615-2e27467e4912%2Fc0e8fa14-f018-4f40-910f-4d60bee085a2%2Fybht3zh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question: Determining Required Scores for Desired Mean in Geography Exams**
Eugene must take four exams in a geography class. If his scores on the first three exams are 91, 67, and 82, what score does he need on the fourth exam for his overall mean to be the following?
(a) at least 70 ______
(b) at least 80 ______
---
**Explanation for Students:**
To solve this problem, let's understand what mean (or average) is first. The mean of a set of numbers is the sum of those numbers divided by the quantity of numbers.
Here's a step-by-step method to solve this:
1. Sum of the first three exam scores:
\[ 91 + 67 + 82 = 240 \]
2. Let the fourth exam score be denoted as \( X \).
3. For part (a), we need the mean of all four exams to be at least 70.
4. Write down the equation for the mean of four exams:
\[ \text{Mean} = \frac{\text{Sum of 4 exam scores}}{\text{Number of exams}} \]
\[ 70 \leq \frac{240 + X}{4} \]
5. Multiply both sides by 4 to clear the fraction:
\[ 70 \times 4 \leq 240 + X \]
\[ 280 \leq 240 + X \]
6. Solve for \( X \):
\[ X \geq 280 - 240 \]
\[ X \geq 40 \]
So, Eugene needs at least a score of 40 on the fourth exam to have an average of at least 70.
Next, let's work on part (b):
1. For part (b), we need the mean of all four exams to be at least 80.
2. Use the same step-by-step method to write down the equation:
\[ 80 \leq \frac{240 + X}{4} \]
3. Multiply both sides by 4:
\[ 80 \times 4 \leq 240 + X \]
\[ 320 \leq 240 + X \]
4. Solve for \( X \):
\[ X \geq 320 - 240 \]
\[ X \geq 80 \]
So, Eugene needs at least a score of 80 on the fourth exam to have an average of at least 80
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